Wireless communication receivers estimate propagation channel characteristics and use the estimates to compensate received signals for channel-induced distortion. More advanced receiver types base interference suppression processing on accurate channel estimation. However, generating accurate channel estimates is challenging, particularly with the growing complexity of communication signal structures.
Multiple-Input-Multiple-Output (MIMO) systems, for example, pose particular challenges, where channel estimation generally must account for the interplay between Ntx transmit antennas and Nrx receive antennas. With pilot-assisted channel estimation, the transmitter transmits a number of known (or pre-determined) symbols from each transmit antenna, thereby allowing estimation of the MIMO channel by the receiver. The LTE standards, as developed by the Third Generation Partnership Project (3GPP), use pilot-assisted channel estimation.
LTE uses an Orthogonal Frequency Division Multiplex (OFDM) carrier signal comprising a plurality of narrowband sub-carriers spanning an overall OFDM bandwidth. Resource allocations assign particular frequencies (sub-carriers) at particular times. In this respect, an OFDM signal “chunk” may be defined as a block of Nt consecutive OFDM symbols (along the time axis) and Nf consecutive sub-carriers (along the frequency axis).
A simplifying assumption is that the channel does not change in time over one chunk and therefore all the pilot symbols are placed in the first OFDM symbols of the chunk. Let {Pj(fm)}m=1M denote the subset of elements of a chunk transmitted from transmit antenna j that are devoted to pilots. That is, M pilot symbols will be transmitted during each chunk from transmit antenna j. The subset of indexes {fm}m=1M for each transmit antenna is determined by the chosen pilot pattern in the frequency-time domain. Similarly, let {Yi,j(fm)}m=1M denote the received signal at the i-th receive antenna corresponding to the pilots {Pj(fm)}m=1M.
Assuming that the pilot symbols transmitted by different antennas are orthogonal, i.e. if Pj(fm) is a pilot symbol on the j-th antenna, then Pj1(fm)=0 for all j1≠j. This implies that the relationship between Yi,j(fm) and {Pj(fm)}m=1M can be described by the following expression:Yi,j(fm)=Hi,j(fm)×Pj(fm)+Vi(fm),1≦m≦M,  (Eq. 1)where Hi,j(f) is the frequency response of the channel between the j-th transmit antenna and the i-th receive antenna corresponding to the f-th sub-carrier, and Vi(f) is a spatially uncorrelated white noise at the i-th receive antenna (antenna thermal noise+other-cell interference) with spectral density gi.
The goal of the channel estimation is to find the estimate of the MIMO channels Hi,j(f) based on observations of {Yi,j(fm)}m=1M and a priori knowledge of the transmitted pilot symbols {Pj(fm)}m=1M=1. One approach is to use Maximum A Posteriori (MAP) channel estimation. Assuming that MIMO channels have Gaussian distribution, it has been shown that MAP channel estimation algorithm can be expressed as
                                                                        H                ⊔                                            i                ,                j                                      ⁡                          (              f              )                                =                                    ∑                              m                =                1                            M                        ⁢                                                            W                  j                                ⁡                                  (                                      f                    ,                                          f                      m                                                        )                                            ⁢                                                Y                                      i                    ,                    j                                                  ⁡                                  (                                      f                    m                                    )                                                                    ,                            (                  Eq          .                                          ⁢          2                )            where i,j(f) denotes the estimate of the channel Hi,j(f).
In Eq. (2), the coefficients Wj(f,fm) are computed as follows:
                                                        W              j                        ⁡                          (                              f                ,                                  f                  m                                            )                                =                                    ∑                              m                =                1                            M                        ⁢                                                            K                  H                                ⁡                                  (                                      f                    ,                                          f                      p                                                        )                                            ⁢                                                P                  j                                ⁡                                  (                                      f                    p                                    )                                            ⁢                                                A                  j                                      -                    1                                                  ⁡                                  (                                                            f                      p                                        ,                                          f                      m                                                        )                                                                    ,                            (                  Eq          .                                          ⁢          3                )            where Aj−1(fp,fm) are the elements of the matrix Aj−1 which is inverse to the matrix Aj with elementsAj(fp,fm)=giδ(fp−fm)+Pj*(fp)KH(fp,fm)Pj(fm),  (Eq. 4)andKH(fp,fm)=E{Hi,j(fp)Hi,j*(fm)}  (Eq. 5)is the correlation matrix of the channel Hi,j(f) in frequency domain. From these expressions, one sees that the MAP-based approach relies on knowledge of second-order channel statistics, including the channel correlation matrix KH(fp,fm), and the noise spectral density gi.
Another well known approach to channel estimation relies on the Maximum Likelihood (ML) algorithm. Denoting the impulse response of the channel between the j-th transmit antenna and the i-th receive antenna by
                                                        h                              i                ,                j                                            (                ML                )                                      ⁡                          (              l              )                                =                                    ∑                              f                =                1                                            N                f                                      ⁢                                                            H                                      i                    ,                    j                                                  ⁡                                  (                  f                  )                                            ⁢              exp              ⁢                              {                                  j2π                  ⁢                                      lf                                          N                      f                                                                      }                                                    ,                  1          ≤          l          ≤          L                ,                            (                  Eq          .                                          ⁢          6                )            the ML channel estimation algorithm can be expressed as
                                                                        h                ^                                            i                ,                j                                      ⁡                          (              l              )                                =                                    ∑                              s                =                1                            L                        ⁢                                                            F                  j                                      -                    1                                                  ⁡                                  (                                      l                    ,                    s                                    )                                            ⁢                                                ∑                                      n                    =                    1                                    N                                ⁢                                                                            p                      j                      *                                        ⁡                                          (                                              s                        -                        n                                            )                                                        ⁢                                                            y                                              i                        ,                        j                                                              ⁡                                          (                      n                      )                                                                                                          ,                            (                  Eq          .                                          ⁢          7                )            where L is the number of channel taps, N is the number of received samples in time domain, yi,j(n) and Pj(n) are respectively Fourier transforms of the received signal Yi,j(f) and pilots Pj(f) at time n*Δt (Δt is a sampling interval), and Fj−1(l,s) are the elements of the matrix Fj−1 which is inverse to the matrix Fj with elements
                                          F            j                    ⁡                      (                          l              ,              s                        )                          =                              ∑                          n              =              1                        N                    ⁢                                                    p                j                *                            ⁡                              (                                  l                  -                  n                                )                                      ⁢                                                            p                  j                                ⁡                                  (                                      s                    -                    n                                    )                                            .                                                          (                  Eq          .                                          ⁢          8                )            
While the ML estimator is simpler to implement in some respects than MAP-based estimators—e.g., ML estimation does not require a priori knowledge of channel statistics, as does MAP estimation—ML estimation can yield poor results in some circumstances. For example, ML estimation does not perform particularly well for MIMO systems with spatially distributed antennas.